You can draw a weapon as a free action combined with "regular move." You can tumble past opponent as part of "normal move." I don't really know if regular and normal moves are one thing or two different things. And in any case, what counts as "normal" or "regular" movement?

Is a charge a "regular move" that lets a barbarian draw his axe as part of it? Can the same barbarian draw his axe as part of climbing a cliff? Can a rogue tumble as part of withdrawing or charging, or while moving through difficult terrain? Can a human tumble while swimming? Can a fish?

2 Answers
2

Normal Movement: This is not a standardized game term, but the rules (and this article) use it to indicate times when a creature uses a move action to move up to its speed -- as opposed to running, charging, or performing some other kind of special movement.

If a creature has a swim, climb, burrow, or fly speed, than those types of movement are considered normal for them. This is slightly extrapolating the not well defined meaning of "normal movement", but part three of the same article discusses this to some extent:

A creature with only a land speed can climb or swim by making an appropriate skill check, but cannot charge, run, withdraw or take a 5-foot step while doing so. That's the chief difference between climbing or swimming with a climb speed and using the Climb or Swim skill.

Since there are a set of heavy restrictions on what you can do with your movement mode, I wouldn't consider it to be "normal" movement at that point. For such a creature, it is a type of "special movement".

So a barbarian couldn't draw an axe while climbing, nor could a rogue tumble while swimming. But creatures with climb or swim speeds could! (Well, you'd still need a free limb somehow for drawing a weapon while climbing.)

NOTE: A few updates have been made to address some of the concerns brought before.
Some of these notes will concern the "Rules Compendium" which not all players use so I'll mark those as such while trying to keep it as legible as possible.

Characters generally don’t walk during combat—they hustle or run.
A Regular move is how you move "tactically" in combat. It consists of using one of your movement abilities for up to it's total'd speed (base speed+bonus speed-armor & other penalties). (Also, notice I said "one of your movement abilities" because sometimes players can have a fly/swim speed which will interact differently with the environment)
Additionally, difficult terrain (loose rocks, mud/grease, vegetation, spells etc) can hamper this movement costing 5 or more feet per square (when using land-speed). Ofcourse, you can always use a 2nd move action inplace of your standard action. This is commonly referred to as a "Double Move Action".

In this case, regular & normal can be used interchangeably. However, certain things can be done as "Part of" a move and others can only be done "as a move" effectively taking the entire move action. The ones you've listed are both "part of move" and Full-Round Actions that take the place of a move.

Charge is something that is technically a full round action and takes up the movement

Withdrawing is also technically a full round action and takes up the movement as well.(or a standard action)

For your direct questions:

Can you draw a weapon as part of a charge?Yes/No/Maybe As has been pointed out, using the PHB, it states "...combined with a regular move (hence the ?) and has been pointed out, Charge is a "Special Full Round Action" not a "regular move action". HOWEVER, the Rules Compendium, drops that requirement and states "You can combine this action with moving your speed" at which point since the charge rules do not explicitly prohibit & a free action can be taken anytime (on your turn) that it isn't prohibited or you haven't already used a free or immediate action. Charge has several stipulations, but not using a free action isn't one of them. So, while it is possible an individual DM's rule may vary, per RAW, it is allowed.

Can you tumble while charging?YES & NO, From SRD: "You must have
a clear path toward the opponent, and nothing can hinder your
movement" HOWEVER, the Rules Compendium (link to SE? regarding RC vs PHB) explicitly states you can Jump, Tumble & make Balance Checks while charging!

Can you tumble while withdrawing?YES, again this is full
round action that lets you move UP-TO twice your move speed (you can
stop earlier if you desire). Obviously, you get no benefit tumbling
from your 1st square, but subsequent squares will provoke
attacks-of-opportunities if they are threatened; tumbling here would
be smart, and using extra movement while moving double your speed is
usually significatnly less of a concern than normal.

Can you tumble while going through difficult terrain?YES,
Difficult Terrain is specifically covered under tumble and
increases the DC accordingly.

Can you tumble while swimming?Nothing in the rules explicitly says yes or no, However, by taking a look at all the rules I believe that, for normal PC Races (those lacking a swim speed) the answer is No! You can not tumble under water. This answer is derived from the entry regarding Deep Bogs. A Deep Bog is 4' deep and impossible to tumble through. (I'm afraid I don't know of any source that more directly addresses this issue in 3.5) However, the RC gives again gives us just a little more to justify a "NO"(for normal races), stating that merely moving through deep, nonﬂowing water requires a natural swim speed or successful Swim checks. For races with a swim speed I think it is perfectly acceptable, by rules and by definition, that they be able to "tumble" in water, although it would most likely differ from us landlubber's would expect it to look like.

5a. NOTE: You can use a tumble or swim check to dive into water and avoid falling damage!

I Forgot your climb question! Sorry! Can you draw an axe as part of a climb? I am changing this one from a "Black & White" answer to a gray one based on As Starwed input. Technically, YES this use is allowed if the character wanting to do so has the right circumstances, which for the most part REQUIRE extra hands. For normal race characters without magical assistance this answer would be NO as the climb rules clearly state "You need both hands to climb", therefore you would have to stop climbing to draw your weapon, which would then be a separate action. Now there are spells that add hands or make your shield animated or weapon dance and Boccob only knows how many other spells that might aide in this function.

6a. (that does it, my next campaign, I'm giving out a +1 dancing climbers pickaxe that will wield itself for 10 rnds/day while climbing! The "weilder" will still be considered armed so he does not provoke AoO while climbing!) sure, pretty use specific, but whatdya think? Sound like a good weapon?

I Have made several edits that substantially changed several of my answers. These edits were mostly made due to the Rules Compendium which not all players or DMs use. So while I personally hate the answer, "ask your DM" some of them you'll have no choice; if you are the DM in question, just decide which books have priority or are allowed and rule accordingly.